I have seen this two successive parenthesis notation in several books but I am still not sure if it is a correct way to express such a non-existence. I also couldn't be sure $\neg \exists$ is the correct way to express "there doesn't exist". Would love to hear ideas on how to express this in a better way, especially if the way I wrote is wrong.
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It's ok. $\phantom{}$ – Canis Lupus Sep 29 '16 at 18:56
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1Your way is correct but you could also write it as, $\forall m, c_m\geq 0$ – ultrainstinct Sep 29 '16 at 19:06
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1It’s fine. I personally prefer $\neg\exists m,(c_m<0)$, and you’ll also see $\neg(\exists M),c_m<0$. – Brian M. Scott Sep 29 '16 at 19:31
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Yes, this is perfectly fine and correct.
Perhaps the more natural way to put the parentheses would be $$ \lnot (\exists m \;:\; c_m < 0). $$ that is, the not ($\lnot$) has the outermost scope, and modifies the entire sentence, not just the $\exists$. But I do not think the above is very common in practice.
As has also been pointed out, you can simplify it by writing $$ \forall m \; : \; c_m \ge 0. $$
Caleb Stanford
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Although I do see it occasionally, I consider that use of the colon non-standard. – Brian M. Scott Sep 29 '16 at 19:34